We prove that Sobolev embeddings can be improved in the presence of symmetries. This includes embeddings in higher

${L}^{p}$-spaces and compactness properties of these embeddings. While such phenomena have been observed in specific context by several authors, we treat here the case of arbitrary Riemannian manifolds (where, in particular, no global chart exist). On the one hand, it turns out that when dealing with compact manifolds, one just has to consider the minimum orbit dimension of the group acting on the manifold. On the other hand, and when dealing with non compact manifolds, one also has to consider the action of the group at infinity. Complete answers are given.